TY - GEN

T1 - Characteristic, Counting, and Representation Functions Characterized

AU - Helou, Charles

PY - 2017/1/1

Y1 - 2017/1/1

N2 - Given a set A of natural numbers, i.e., nonnegative integers, there are three distinctive functions attached to it, each of which completely determines A. These are the characteristic function χA(n) which is equal to 1 or 0 according as the natural number n lies or does not lie in A, the counting function A(n) which gives the number of elements a of A satisfying a ≤ n, and the representation function rA(n) which counts the ordered pairs (a, b) of elements a, b ∈ A such that a + b = n. We establish direct relations between these three functions. In particular, we express each one of them in terms of each other one. We also characterize the representation functions by an intrinsic recursive relation which is a necessary and sufficient condition.

AB - Given a set A of natural numbers, i.e., nonnegative integers, there are three distinctive functions attached to it, each of which completely determines A. These are the characteristic function χA(n) which is equal to 1 or 0 according as the natural number n lies or does not lie in A, the counting function A(n) which gives the number of elements a of A satisfying a ≤ n, and the representation function rA(n) which counts the ordered pairs (a, b) of elements a, b ∈ A such that a + b = n. We establish direct relations between these three functions. In particular, we express each one of them in terms of each other one. We also characterize the representation functions by an intrinsic recursive relation which is a necessary and sufficient condition.

UR - http://www.scopus.com/inward/record.url?scp=85042644184&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85042644184&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-68032-3_9

DO - 10.1007/978-3-319-68032-3_9

M3 - Conference contribution

AN - SCOPUS:85042644184

SN - 9783319680309

T3 - Springer Proceedings in Mathematics and Statistics

SP - 139

EP - 155

BT - Combinatorial and Additive Number Theory II - CANT

A2 - Nathanson, Melvyn B.

PB - Springer New York LLC

T2 - 13th Workshop on Combinatorial and Additive Number Theory, CANT 2015

Y2 - 19 May 2015 through 22 May 2015

ER -